A209329 Decimal expansion of the sum over the inverse products of adjacent odd primes.

1, 3, 4, 4, 2, 6, 5, 0, 9, 6, 9, 1, 7, 3, 3, 2, 2, 8

Offset

0,2

Comments

Contains the contribution from twin primes (A209328) plus other contributions from cousin primes (A143206) not already part of twin primes, sexy primes (A210477) not already accounted for, etc.

Summing up to (and including) 12-digit primes yields 0.134426509691698261. - Hans Havermann, Mar 17 2013

Links

Table of n, a(n) for n=0..17.

R. J. Mathar, Estimate of the Sum over Inverse Products of Adjacent Prime Pairs

Formula

sum_{3 < p < 10^4} 1/(prevprime(p)*p) = 0.134416688[9]...

sum_{3 < p < 10^5} 1/(prevprime(p)*p) = 0.134425707...

sum_{3 < p < 10^6} 1/(prevprime(p)*p) = 0.1344264419...

sum_{3 < p < 10^7} 1/(prevprime(p)*p) = 0.13442650383...

sum_{3 < p < 10^8} 1/(prevprime(p)*p) = 0.13442650917[5]...

sum_{3 < p < 10^9} 1/(prevprime(p)*p) = 0.13442650964545...

Extrapolation of this data (using Aitken's method) indeed suggests a value of 0.134426509692, rounded to the last decimal place. Extrapolation of the ratios of the first differences (9.02e-6, 7.35e-7, 6.19e-8, 5.34e-9, 4.699e-10) yields subsequent terms (4.26e-11, 4.0e-12). - M. F. Hasler, Jan 22 2013

Example

0.134426509... = 1/(3*5) + 1/(5*7) + 1/(7*11) + 1/(11*13)+ ... = Sum_{n>=2} 1/A006094(n).

Prog

(PARI) {default(realprecision, 19); s=0; q=1/3; forprime(p=1/q+1, 10^9, s+=q*q=1./p); s} /* M. F. Hasler, Jan 22 2013 */

Crossrefs

Cf. A210473 (includes 1/(2*3)). Cf. also A085548.

Sequence in context: A103121 A291086 A358449 * A353156 A106290 A249618

Adjacent sequences: A209326 A209327 A209328 * A209330 A209331 A209332

Keyword

cons,nonn,more

Author

R. J. Mathar, Jan 19 2013

Extensions

More terms from R. J. Mathar, Feb 08 2013

Status

approved